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Mathematical Engineering Dieter Gerling Electrical Machines Mathematical Fundamentals of Machine Topologies Electrical Machines Mathematical Engineering Series Editors Prof. Dr. Claus Hillermeier, Munich, Germany, (volume editor) Prof. Dr.-Ing. Jörg Schröder, Essen, Germany Prof. Dr.-Ing. Bernhard Weigand, Stuttgart, Germany For furthervolumes: http://www.springer.com/series/8445 Dieter Gerling Electrical Machines Mathematical Fundamentals of Machine Topologies Dieter Gerling Fakultät Elektrotechnik und Informationstechnik Universität der Bundeswehr München Neubiberg, Germany ISSN 2192-4732 ISSN2192-4740 (electronic) ISBN 978-3-642-17583-1 ISBN 978-3-642-17584-8 (eBook) DOI 10.1007/978-3-642-17584-8 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2014950816 © Springer-Verlag Berlin Heidelberg 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Calculation and design of electrical machines and drives remain challenging tasks. However, this becomes even more and more important as there are increasing numbers of applications being equipped with electrical machines. Some recent examples well-known to the public are wind energy generators and electrical trac- tion drives in the automotive industry. To realize optimal solutions for electrical drive systems it is necessary not only to know some basic equations for machine calculation, but also to deeply understand the principles and limitations of electrical machines and drives. To foster the know-how in this technical field, this book Electrical Machines starts with some basic considerations to introduce the reader to electromagnetic circuit calculation. This is followed by the description of the steady-state operation of the most important machine topologies and afterwards by the dynamic operation and control methods. Continuously giving detailed mathematical deductions to all topics guarantees an optimal understanding of the underlying principles. Therefore, this book contributes to a comprehensive expert knowledge in electrical machines and drives. Consequently, it will be very useful for academia as well as for industry by supporting senior students and engineers in conceiving and de- signing electrical machines and drives. After introducing Maxwell’s equations and some principles of electromagnetic circuit calculation, the first part of the book is dedicated to the steady-state operation of electrical machines. The detailed description of the brushed DC-machine is followed by the rotating field theory, which in particular explains in detail the winding factors and harmonics of the magneto-motive force of distributed windings. On this basis, induction machines and synchronous machines are described. This first part of the book is completed by regarding permanent magnet machines, switched reluctance machines, and small machines for single-phase use. Dynamic operation and control of electrical machines are the topics of the second part of this book, starting with some fundamental considerations. Next, the dynamic operation of brushed DC-machines and their control is described (in particular cascaded control using PI-controllers and their adjustment rules). A very important concept for calculating the dynamic operation of rotating field machines is the space vector theory; this is deduced and explained in detail in the following chapter. Then, the dynamic behavior of induction machines and synchronous machines follows, including the description of important control methods like field- v vi Preface oriented control (FOC) and direct torque control (DTC). The permanent magnet machine with surface mounted magnets (SPM) or interior magnets (IPM) is explained concerning the differences of both in torque control and concerning the maximum torque per ampere (MTPA) control method. The last chapter gives an overview of latest research results concerning concentrated windings. In spite of being a new contribution to a comprehensive understanding of electrical machines and the respective actual developments the reader may find parts of the contents even in different literature, as this book explains the fundamentals of electrical machines (steady-state and dynamic operation as well as control). Concerning these fundamentals it is nearly impossible to list all relevant literature during the text layout. Therefore, the most important references are given at the end of each chapter. In addition, parts of the lectures of Prof. H. Bausch (Universi- taet der Bundeswehr Munich, Germany) and Prof. G. Henneberger (RWTH Aa- chen, Germany) were used as a basis. The author deeply wishes to express his grateful acknowledgment to all team members of his Chair of Electrical Drives and Actuators at the Universitaet der Bundeswehr Munich and of the spin-off company FEAAM GmbH for their most valuable discussions and support. In particular this holds for (in alphabetical or- der) Dr.-Ing. Gurakuq Dajaku, Mrs. Lara Kauke, and most notably Dr.-Ing. Hans- Joachim Koebler. Without their beneficial contributions this book would not have been possible in such a high quality. Last, but not least the author exceedingly thanks his wife and his daughters for their respectfulness and understanding not only concerning the effort being ac- companied by writing this book, but even concerning the expenditure of time the author dedicates to professional activities. Munich, April 2014 Dieter Gerling Contents Preface ................................................................................................................... .v(cid:3) Contents ...............................................................................................................v ii(cid:3) 1 Fundamentals ..................................................................................................... 1(cid:3) 1.1 Maxwell’s Equations ................................................................................... 1(cid:3) 1.1.1 The Maxwell’s Equations in Differential Form ................................... 1(cid:3) 1.1.2 The Maxwell’s Equations in Integral Form .......................................... 2(cid:3) 1.1.2.1 Ampere’s Law (First Maxwell’s Equation in Integral Form) ....... 2(cid:3) 1.1.2.2 Faraday’s Law, Law of Induction (Second Maxwell’s Equation in Integral Form) ........................................................................................... 3(cid:3) 1.1.2.3 Law of Direction ........................................................................... 4(cid:3) 1.1.2.4 The Third Maxwell’s Equation in Integral Form .......................... 4(cid:3) 1.1.2.5 The Fourth Maxwell’s Equation in Integral Form ........................ 4(cid:3) 1.1.2.6 Examples for the Ampere’s Law (First Maxwell’s Equation in Integral Form) ........................................................................................... 5(cid:3) 1.1.2.7 Examples for the Faraday’s Law (Second Maxwell’s Equation in Integral Form) ........................................................................................... 7(cid:3) 1.2 Definition of Positive Directions ............................................................... 14(cid:3) 1.3 Energy, Force, Power ................................................................................ 16(cid:3) 1.4 Complex Phasors ....................................................................................... 28(cid:3) 1.5 Star and Delta Connection ......................................................................... 30(cid:3) 1.6 Symmetric Components ............................................................................. 31(cid:3) 1.7 Mutual Inductivity ..................................................................................... 32(cid:3) 1.8 Iron Losses ................................................................................................. 34(cid:3) 1.9 References for Chapter 1 ........................................................................... 35(cid:3) 2 DC-Machines .................................................................................................... 37(cid:3) 2.1 Principle Construction ............................................................................... 37(cid:3) 2.2 Voltage and Torque Generation, Commutation ......................................... 38(cid:3) 2.3 Number of Pole Pairs, Winding Design ..................................................... 41(cid:3) 2.4 Main Equations of the DC-Machine .......................................................... 45(cid:3) 2.4.1 First Main Equation: Induced Voltage ............................................... 45(cid:3) 2.4.2 Second Main Equation: Torque .......................................................... 47(cid:3) 2.4.3 Third Main Equation: Terminal Voltage ............................................ 48(cid:3) vii viii Contents 2.4.4 Power Balance .................................................................................... 48(cid:3) 2.4.5 Utilization Factor ............................................................................... 49(cid:3) 2.5 Induced Voltage and Torque, Precise Consideration ................................. 51(cid:3) 2.5.1 Induced Voltage ................................................................................. 51(cid:3) 2.5.2 Torque ................................................................................................ 53(cid:3) 2.6 Separately Excited DC-Machines .............................................................. 56(cid:3) 2.7 Permanent Magnet Excited DC-Machines ................................................. 62(cid:3) 2.8 Shunt-Wound DC-Machines ...................................................................... 69(cid:3) 2.9 Series-Wound DC-Machines ..................................................................... 73(cid:3) 2.10 Compound DC-Machines ........................................................................ 77(cid:3) 2.11 Generation of a Variable Terminal Voltage ............................................. 78(cid:3) 2.12 Armature Reaction ................................................................................... 80(cid:3) 2.13 Commutation Pole ................................................................................... 84(cid:3) 2.14 References for Chapter 2 ......................................................................... 88(cid:3) 3 Rotating Field Theory ...................................................................................... 89(cid:3) 3.1 Stator of a Rotating Field Machine ............................................................ 89(cid:3) 3.2 Current Loading ......................................................................................... 90(cid:3) 3.3 Alternating and Rotating Magneto-Motive Force ...................................... 93(cid:3) 3.4 Winding Factor ........................................................................................ 104(cid:3) 3.5 Current Loading and Flux Density........................................................... 114(cid:3) 3.5.1. Fundamentals .................................................................................. 114(cid:3) 3.5.2. Uniformly Distributed Current Loading in a Zone .......................... 114(cid:3) 3.5.3. Current Loading Concentrated in the Middle of Each Slot ............. 115(cid:3) 3.5.4. Current Loading Distributed Across Each Slot Opening ................ 116(cid:3) 3.5.5. Rotating Air-Gap Field .................................................................... 116(cid:3) 3.6 Induced Voltage and Slip ......................................................................... 120(cid:3) 3.7 Torque and Power .................................................................................... 126(cid:3) 3.8 References for Chapter 3 ......................................................................... 134(cid:3) 4 Induction Machines ........................................................................................ 135(cid:3) 4.1 Construction and Equivalent Circuit Diagram ......................................... 135(cid:3) 4.2 Resistances and Inductivities ................................................................... 141(cid:3) 4.2.1 Phase Resistance .............................................................................. 141(cid:3) 4.2.2 Main Inductivity ............................................................................... 142(cid:3) 4.2.3 Leakage Inductivity .......................................................................... 142(cid:3) 4.2.3.1 Harmonic Leakage .................................................................... 142(cid:3) 4.2.3.2 Slot Leakage ............................................................................. 143(cid:3) 4.2.3.3 End Winding Leakage............................................................... 145(cid:3) 4.3 Operating Characteristics ......................................................................... 145(cid:3) 4.3.1 Heyland-Diagram (Stator Phase Current Locus Diagram) ............... 145(cid:3) 4.3.2 Torque and Power ............................................................................ 151(cid:3) 4.3.3 Torque as a Function of Slip ............................................................ 154(cid:3) 4.3.4 Series Resistance in the Rotor Circuit .............................................. 157(cid:3) Contents ix 4.3.5 Operation with Optimum Power Factor ........................................... 159(cid:3) 4.3.6 Further Equations for Calculating the Torque .................................. 164(cid:3) 4.4 Squirrel Cage Rotor ................................................................................. 166(cid:3) 4.4.1 Fundamentals ................................................................................... 166(cid:3) 4.4.2 Skewed Rotor Slots .......................................................................... 170(cid:3) 4.4.3 Skin Effect ........................................................................................ 175(cid:3) 4.5 Possibilities for Open-Loop Speed Control ............................................. 179(cid:3) 4.5.1 Changing (Increasing) the Slip ......................................................... 179(cid:3) 4.5.2 Changing the Supply Frequency ...................................................... 179(cid:3) 4.5.3 Changing the Number of Pole Pairs ................................................. 181(cid:3) 4.6 Star-Delta-Switching ............................................................................... 182(cid:3) 4.7 Doubly-Fed Induction Machine ............................................................... 183(cid:3) 4.8 References for Chapter 4 ......................................................................... 188(cid:3) 5 Synchronous Machines .................................................................................. 189(cid:3) 5.1 Equivalent Circuit and Phasor Diagram................................................... 189(cid:3) 5.2 Types of Construction .............................................................................. 195(cid:3) 5.2.1 Overview .......................................................................................... 195(cid:3) 5.2.2 High-Speed Generator with Cylindrical Rotor ................................. 196(cid:3) 5.2.3 Salient-Pole Generator ..................................................................... 196(cid:3) 5.3 Operation at Fixed Mains Supply ............................................................ 196(cid:3) 5.3.1 Switching to the Mains ..................................................................... 196(cid:3) 5.3.2 Torque Generation............................................................................ 198(cid:3) 5.3.3 Operating Areas ............................................................................... 200(cid:3) 5.3.4 Operating Limits .............................................................................. 203(cid:3) 5.4 Isolated Operation .................................................................................... 205(cid:3) 5.4.1 Load Characteristics ......................................................................... 205(cid:3) 5.4.2 Control Characteristics ..................................................................... 207(cid:3) 5.5 Salient-Pole Synchronous Machines ........................................................ 209(cid:3) 5.6 References for Chapter 5 ......................................................................... 217(cid:3) 6 Permanent Magnet Excited Rotating Field Machines................................. 219(cid:3) 6.1 Rotor Construction ................................................................................... 219(cid:3) 6.2 Linestart-Motor ........................................................................................ 220(cid:3) 6.3 Electronically Commutated Rotating Field Machine with Surface Mounted Magnets ......................................................................................................... 220(cid:3) 6.3.1 Fundamentals ................................................................................... 220(cid:3) 6.3.2 Brushless DC-Motor ........................................................................ 222(cid:3) 6.3.3 Electronically Commutated Permanent Magnet Excited Synchronous Machine ..................................................................................................... 228(cid:3) 6.4 Calculation of the Operational Characteristics; Permanent Magnet Excited Machines with Buried Magnets ..................................................................... 230(cid:3) 6.5 References for Chapter 6 ......................................................................... 230(cid:3)